Please give me a reference to a book or lecture notes where the following stuff is studied.

Let $M$ be a Riemann surface with boundary $\partial M$ (but not necessarily, any smooth $n$-dimensional manifold suffices if the following statements make sense in this case).

  1. What does it mean for $1$-form $\omega$ on $M$ to have $\int\limits_\gamma \omega = 0$ $(\operatorname{mod} 2\pi)$ for any closed loop $\gamma$?
  2. If $\omega_1$, $\ldots$, $\omega_N$ are closed forms which form a basis of $H^1(M)$ (de-Rham cohomology) then there are non-homotopically equivalent loops $\gamma_1$, $\ldots$, $\gamma_N$ in $M$ such that $\int_{\gamma_i} \omega_j = \delta_{ij}$. How to prove this? Is it possible to say more about these $\gamma_1$, $\ldots$, $\gamma_N$? Do they have some interpretation on terms of $H_1(M)$ (singular homology)?
  3. As in 2., if $\omega_1$, $\ldots$, $\omega_N$ are closed 1-forms which form a basis of $H^1(M,\partial M)$ there exist non-homotopically equivalent loops $\gamma_1$, $\ldots$, $\gamma_N$, non-homotopically equivalent to any component of $\partial M$, such that $\int_{\gamma_i} \omega_j = \delta_{ij}$. What can we say about these curves in comparison with question 2?
  • 1
    $\begingroup$ I think that do Carmo's book "differential forms and applications" will be helpful for problem 1 $\endgroup$ – HK Lee Nov 26 '13 at 10:51
  • $\begingroup$ @HeeKwonLee unfortunately I haven't found the answer in this book, do you know the answer? $\endgroup$ – Appliqué Nov 26 '13 at 17:05
  • $\begingroup$ Sorry i did not read condition well $\endgroup$ – HK Lee Nov 27 '13 at 1:07

What you are struggling with is called Poincare Duality for manifolds with boundary. If $M$ is a compact oriented $n$-dimensional manifold (in your case, $n=2$, $k=1$), the Poincare duality reads: $$ H^k(M)\cong H_{n-k}(M, \partial M), H_k(M)\cong H^{n-k}(M, \partial M). $$ In the case $n=2$, you also have the Kronecker duality $$ H^k(M)\cong (H_k(M))^*, H^k(M, \partial)\cong (H_k(M,\partial))^* $$ (in general, you have to use real coefficients or use the Universal Coefficients Theorem, which is more subtle). The best place to read about this that I know is the book

R. Bott, L. Tu, "Differential forms in algebraic topology".


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.